Substituting values in a Python sympy unknown function

Question

I want to solve an implicit functional equation by finding the solution's Taylor expansion. E.g., solve

n f(x) - (n-1) f(f(x)) - f'(f(x)) f(x) = x^(n-1) f(x)

for f(x), given f(0) = 0. Differentiating this with respect to x, I get

n f'(x) - n f'(f(x)) f'(x) - f''(f(x)) f(x) f'(x) = (n-1) x^(n-2) f(x) + x^(n-1) f'(x)

So, for n > 1: n - n f'(0) = 0, i.e., f'(0) = 1

I want to derive f''(0), f'''(0), etc., similarly by differentiating and substituting in sympy (or other Python package). How to do it?

It's not clear to me how to tell sympy that f(0)=0? or that f'(0)=1? etc.